As mentioned in more details at HELP.gv.at (see previous post), persons establishing their residence in Austria, changing residence within Austria, or changing their primary residence must register by filling out and submitting a Residence Registration Form (Meldezettel) to the responsible authority. In Vienna, you need to go to the Municipal Authority (Magistrat) of your district.

The documents to be presented (based on Help.gv.at) are (1) Travel document (e.g. passport), (2) Birth certificate, (3) Residence Registration Form (Meldezettel), and (4) Documentary Proof of Academic Degrees. It seems that, in reality, only the Meldezettel and the Passport are needed.